High-power electrically pumped terahertz topological laser based on a surface metallic Dirac-vortex cavity

Topological lasers (TLs) have attracted widespread attention due to their mode robustness against perturbations or defects. Among them, electrically pumped TLs have gained extensive research interest due to their advantages of compact size and easy integration. Nevertheless, limited studies on electrically pumped TLs have been reported in the terahertz (THz) and telecom wavelength ranges with relatively low output powers, causing a wide gap between practical applications. Here, we introduce a surface metallic Dirac-vortex cavity (SMDC) design to solve the difficulty of increasing power for electrically pumped TLs in the THz spectral range. Due to the strong coupling between the SMDC and the active region, robust 2D topological defect lasing modes are obtained. More importantly, enough gain and large radiative efficiency provided by the SMDC bring in the increase of the output power to a maximum peak power of 150 mW which demonstrates the practical application potential of electrically pumped TLs.

For the two-dimensional system, the effective continuum Hamiltonian takes the Firstly, we calculate the effective refractive index of the device with non-etched active region for later calculations of the band diagram of the topological cavity based on the finite element method.As the thickness of the active region increases, the effective refractive index increases rapidly.The thickness of our active region is 11.7 μm.As shown in Fig. S1b, the effective refractive indexes of 3.08 and 3.6 are obtained for structures without and with metal covering, respectively.A large effective refractive index contrast can be achieved for a thinner active region layer.Although the decrease in the thickness of the active region will introduce reduced cascade periods and thus smaller gain, the large refractive index contrast can improve the surface emission efficiency, which may also result in high output power.S3.As is show in Fig. S3, the absorption boundary design has played an important role in the realization of highly robust single-mode lasing.The device without absorption boundary can't operate stably in topological mode while the device with absorption boundary demonstrates the robust operation of topological mode, which is illustrated by both the lasing spectra and the far-field patterns.This can be attributed to the increasing loss of the whispering gallery modes due to the introduction of the absorption boundary.Besides, in Fig. S3c and S3f, the circle mesa cavity with circular absorption boundary device demonstrates multimode operation and the far-field also does not satisfy the C3v symmetry, since the whispering-gallery mode will have a lower loss for a circular boundary.As is shown in Fig. S4a, although the second-order DFB device demonstrates single-mode operation, the far-field demonstrates a long strip pattern.To maintain the fundamental mode operation, the ridge width of the second-order DFB device is limited which results in a large far-field divergence angle.The device with  = 0 demonstrates multi-mode operation.This can be attributed to the unoptimized design of this photonic crystal structure, which results in a band-edge structure with many multiple modes having the similar gain condition for lasing.However, from another perspective, this demonstrates the important role of the introduction of the vortex cavity for the single-mode operation.The photonic crystals device can also realize a large-threshold gain difference with carefully design 3,4 .Compare with photonic crystal device lasing on the band edge modes, the topological cavity device introduces an approach to obtain stable mid-gap single-mode operation, which gives a large FSR and threshold margin simultaneously.

S6. Calculation of loss and vertical radiative efficiency of the cavity modes
The loss of the cavity modes and vertical radiative efficiency are calculated using the three-dimensional (3D, full-wave finite element method.For the calculations, we have taken the device with  = 0.18 and  0 = 30.5μmas the example.
The calculated Q factors for different modes are shown in Figure S5.The photon loss rate  r due to the vertical radiation of the laser is estimated by extracting the timeaveraged integrated power flow through the open air domains and normalizing it with respect to the resonator energy in the 3D simulation 5 : Where E and H represent the electric and magnetic fields respectively,  ̂ represents the unit normal vector of the circle air domains,  the dielectric constant and  the permeability.The corresponding quality factors have been derived from the relation   = /  , where  is the eigenfrequency of the topological mode.
And the radiative out-coupling efficiency   is assumed to be proportional to where: Based on the 3D simulation, a photon loss rate of   ≈ 28.5 GHz from integral results through equation (R.1, is obtained, which corresponds to   = 140 . Considering the calculated   = 96 from the 3D simulation, an in-plane quality factor   = 320 is obtained.As a result, an out-coupling efficiency of   ≈ 68.5% is estimated.This result is obtained without considering the waveguide loss.) , and the facet reflectivity for metal-metal waveguide structure with waveguide width of 100 μm and active region thickness of 11.7 μm around 3.2 THz is about 80% 6 .Therefore we obtain   ≈ 1.1 −1 .
Meanwhile, the total optical loss coefficient   for F-P plasmonic QCLs operating in the wavelength range of 3~4 THz [6][7][8] has been experimentally measured in the range of 10~15 −1 .Considering   =   +   , where   is waveguide loss, we take the   = 12.5 −1 and   = 11.4−1 is obtained.Figure S6 shows the L-I-V curves of the F-P device with slope efficiency of 28 /.In theory, the slope efficiency of the F-P device can be written as Meanwhile, as for the topological cavity device, the total optical loss coefficient and the slope efficiency can be written as Where  ⊥ represent optical loss coefficient due to the vertical radiation,  ∥ is the inplane mirror loss which can be obtained from α =  S1 compared our results with other typical high-power surface-emitting THz QCL reported previously.Clearly, our device exhibits fairly competitive radiation loss and efficiency compared with devices based on different photonic designs, providing the huge potential for high output power.For the absolute output power of these devices, the epitaxial material used in the device fabrications is another critical factor, which may introduce the inconsistency between the radiation efficiency and output power.Therefore, enough gain provided by the surface metal cavity design and large vertical radiation loss and efficiency introduced by the topological photonics design result in the increase of the output power of our device together.(1) Estimated with the typical parameters from ref [6-8]   (2) Calculated from the Q factor As shown in Fig. S7, the output power of the device with  of 14 0 is nearly twice the power of the device with  of 10 0 , which demonstrates nearly linear power upscaling with increasing sizes.In further, the device can be made much larger for higher output power.Due to the large free spectral range (FSR, of the topological midgap mode, stable single mode operation is expected with larger device size.In further, smaller divergence angle will be obtained increasing device size.However, the biggest challenge for large-area device is the heat dissipation issue of the device, nevertheless, this issue could be potentially addressed by thinning the thickness of the active region accordingly.As is shown in Fig. S6, the devices with phase modulation can maintain the properties of vortex polarization while the symmetry of the far-field patterns changes from C3 to double-lobed beam ( = 3, and doughnut beam ( = 1.5,.In further, the lasing spectra also remain the same as that of device without phase modulation.These results shows that the symmetry of the output far-field can be tuned through phase modulation with topological lasing properties unchanged.

1 ) 2 )(S1. 3 )
Fig. S1 | Effective refractive index calculation.a Simulation diagram and b Effective refractive index changes as the thickness of the active region.

Fig. S2 |
Fig. S2 | Performance of the SMDC device with  = . and   = .μm under different temperatures.a L-I-V results under different temperatures.b Lasing spectra under different temperatures.c.d.e Far-field patterns under different temperatures.We also characterized the performances of the SMDC TLs under different temperatures, as shown in Fig. S2.With the increasing temperatures from 8.5 K to 35 K, the devices still maintain robust single-mode operations and C3V symmetric farelectric fields, which indicate the operating robustness of the devices with different temperatures.

Fig. S3 |
Fig. S3 | The effects of the absorbing boundary condition on lasing properties.a,d Farfield pattern and lasing spectrum of the hexagonal mesa cavity device without absorbing boundary.b,e Far-field and spectrum of hexagonal mesa cavity device with 15 μm wide highly doped absorbing boundary.c,f Far-field and spectrum of circle mesa device C with 15 μm wide highly doped absorbing boundary.To study the effects of the absorbing boundary condition on lasing properties, we fabricated devices with different boundary condition for comparison.Lasing spectra and far-field patterns of the devices with different boundary conditions and parameter of  = 0.18 and  0 = 30.5 μ m are shown in Fig. S3.As is show in Fig. S3, the

Fig. S4 |
Fig. S4 | Far-field patterns and lasing spectra for different types of devices.a,d Far-field pattern

Figure S5 .
Figure S5.3D simulation of the Q factors for different modes Next, we use the experimental results of the F-P device from the same wafer as the topological cavity device to modify the theoretical calculation considering the waveguide loss.For the F-P device with 2 mm cavity length and 100 μm ridge width, we have   = 1 2 ln ( 1  1  2 ) = 1   ( 1

Figure S6 .
Figure S6.L-I-V curves of the F-P device with 2 mm cavity length and 100 μm ridge width.

. 5 ( 5 = 5 .
Hence  ⊥ (  =140, and  ⊥ +  ∥ (   =66.7, are calculated to be 17.5 −1 and 25.5 −1 respectively.Therefore, after taking the waveguide loss   in to consideration, we obtain  _ = 17.25.5+11.4)= 0.474 for the SMDC TL device with  = 0.18 and  0 = 30.5 μm.Meanwhile, the slope efficiency  _ 151/ is obtained from the P-I-V curve in Fig.3bexperimentally.Therefore, we have the theoretically calculated slope efficiency ratio for the topological and FP devices 386, which is in excellent agreement with the experimental data 393.This verifies the reliability of the numerical simulation.This relatively high outcoupling efficiency originates from the coherent constructive emission over the entire lattice area.Table

Fig. S7 |
Fig. S7 | Power performance of devices with different sizes.

Fig. S8 |
Fig. S8 | Schematic diagram of the far-field testing optical pathAs illustrated in Fig.S8, the far-field properties of the devices were directly measured by a Golay cell detector scanning on the curve of a sphere with a radius around 15 cm from the device.For the polarized properties measurement, one linear polarizer was added to filter the cross-polarized components during the far-field intensity scanning.Through changing the direction of the linear polarizer, the polarized far-field properties were measured.

Table S1 .
Radiation loss and efficiency for different single-chip THz surface emitting QCLs